Problem 41 - Algebra for College Students

Question

A football is on the 8-yard line, and five penalties in a row are given that move the ball half the distance to the (closest) goal. Write a sequence of five terms that specify the location of the ball after each penalty.

Step-by-Step Solution

Answer

The ball's positions after each penalty are 4, 2, 1, 0.5, and 0.25 yards from the goal line.

Step 1: Identify the initial position

The football starts at the 8-yard line.

Step 2: Apply the first penalty

After the first penalty, the ball moves half the distance to the goal line. This is: \[\frac{8}{2} = 4 \text{ yards from the goal line}\]

Step 3: Apply the second penalty

After the second penalty, the ball again moves half the remaining distance. This is: \[\frac{4}{2} = 2 \text{ yards from the goal line}\]

Step 4: Apply the third penalty

After the third penalty, the ball moves half of 2 yards. This is: \[\frac{2}{2} = 1 \text{ yard from the goal line}\]

Step 5: Apply the fourth penalty

After the fourth penalty, the ball moves half of 1 yard. This is: \[\frac{1}{2} = 0.5 \text{ yards from the goal line}\]

Step 6: Apply the fifth penalty

After the fifth penalty, the ball moves half of 0.5 yards. This is: \[\frac{0.5}{2} = 0.25 \text{ yards from the goal line}\]

Key Concepts

Geometric Sequences

In the given problem, the football’s movement can be described through a geometric sequence. A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the common ratio is \(\frac{1}{2}\).
Each term in the sequence is obtained by multiplying the previous position of the football by \(\frac{1}{2}\).
Starting from the initial position of 8 yards, the terms are:
\[\begin{align*} 8 \rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow 0.5 \rightarrow 0.25 \end{align*}\]
Notice how each subsequent term is half of the previous term. This pattern is the hallmark of a geometric sequence.
Understanding geometric sequences helps in predicting how a value changes over iterations, especially when dealing with exponential decay or growth scenarios.

Distance Calculation

Distance calculation involves determining how far the football moves after each penalty. Each penalty moves the ball halfway closer to the goal line. The distance moved can be computed as:
- Initial Distance: 8 yards from the goal line
- After 1st penalty: 8 * \(\frac{1}{2}\) = 4 yards
- After 2nd penalty: 4 * \(\frac{1}{2}\) = 2 yards
- After 3rd penalty: 2 * \(\frac{1}{2}\) = 1 yard
- After 4th penalty: 1 * \(\frac{1}{2}\) = 0.5 yards
- After 5th penalty: 0.5 * \(\frac{1}{2}\) = 0.25 yards
By calculating the distances each time, you observe how the ball is progressively moving closer to the goal line. This type of repetitive halving characterizes a geometric sequence, and helps understand how objects or values change incrementally over time.

Penalty Application

Each penalty applied in the problem affects the position of the football in a predictable manner. The penalties are designed to cut the remaining distance to the goal line in half.
This scenario is mildly different from regular movements because typically, penalties move the ball a fixed distance back rather than halving its current distance.
Here’s the breakdown of penalty applications:

• 1st Penalty: Ball moves from 8 yards to 4 yards.
• 2nd Penalty: Ball moves from 4 yards to 2 yards.
• 3rd Penalty: Ball moves from 2 yards to 1 yard.
• 4th Penalty: Ball moves from 1 yard to 0.5 yards.
• 5th Penalty: Ball moves from 0.5 yards to 0.25 yards.

The consistent halving of the distance each time creates a series of penalties that form a geometric sequence.
Understanding this penalty system helps in various practical scenarios, such as programming algorithms where iterative halving or incremental reductions are needed.